attached greater significance to high frequency magnetic components. As a result
... design packages for these components are required. The basic transformer ...
DESIGN OF MAGNETIC COMPONENTS USING A GRAPHICAL INTERFACE: INCLUSIVE OF HIGH FREQUENCY EFFECTS J.G. Breslin, W.G. Hurley Department of Electronic Engineering, University College, Galway, Ireland E-mail
[email protected],
[email protected] ABSTRACT The movement towards high density electronics has attached greater significance to high frequency magnetic components. As a result, user-friendly computer aided design packages for these components are required. The basic transformer design formulæ are examined, and the conditions for optimising transformer design at high frequencies are established. This paper introduces a high quality graphics package which is currently being developed for the Microsoft Windows environment to implement these design rules for the first time. 1. INTRODUCTION Recently, a trend has emerged towards design at higher frequencies due to the space constraints in modern electronic equipment, and optimising a transformer design at these frequencies has received renewed attention during the past few years. Such design also requires increased attention to the problems of losses and optimisation, and references [1] to [4] discuss among other things the minimisation of skin and proximity effect losses in copper conductors, reduction of core losses for maximum efficiency, the calculation of the optimum point of flux density and also the maximum VA output. A new design package has been developed for the Windows environment to implement some of the techniques proposed in these papers. Since a transformer is only as reliable as the rules which were used to design it, the fundamental design procedure is being examined in detail to eliminate any discrepancies involved in the process. The high frequency effects are of particular interest here as many previous design methods began with the assumption that the core and winding losses are equal whereas in this new design method it has been shown that the core losses increase with frequency much faster than winding losses. Due to the powerful graphical capabilities of the Visual Basic for Windows programming language, it was selected for use in this package. The package will guide the user through a number of fundamental design steps, for example, selecting a core subject to design specifications, then calculating the number of turns required, the wire sizes involved, as well as displaying the core/winding losses and skin/proximity effects.
able to use the package in a very short time with little instruction. The advent of Visual Basic meant that for the first time, Windows programs could be written from within Windows without the enormous overburden of code needed with C. Thus the revolution promised by Object-Oriented Programming (OOP) experts has been subdued somewhat by the relative simplicity with which custom applications can be written in Visual Basic. The traditional OOP vision was that reusable objects would appear as by-products of software development, a market would emerge, and programmers would become producers and consumers of objects. However, this rather vague idea never really came about due to the inability of most OOP languages, including C++, to package and distribute objects effectively in binary form. Visual Basic’s custom controls (VBXes) best exemplify the decades-old notion of reusable software, and this has come as something of a surprise to most programmers since OOP specialists claim that real objects show all the desirable software properties of inheritance, polymorphism and encapsulation, while VBXes only stand on a single leg of encapsulation. Yet despite their claims, VBX - rather than OOP - has ignited the component software revolution. This can be seen in the fact that C++ vendors such as Microsoft and Borland have reverse-engineered their products so that programmers, lacking reusable C++ objects, can tap into the rich VBX component market. 3. DESIGN PACKAGE In order to successfully implement a computer aided design procedure for transformers, the basic design rules must be examined and common ground found for the various transformer applications. The core selection process is usually the first design step involved, and a standard format for entering the transformer specifications is shown in Figure 1, which is a sample screen from the design package. A data flow diagram (DFD) of the process is given in Figure 2.
A number of applications will be described, e.g. push-pull converters and forward converters operating up to 50 kHz. The prediction of losses for non-sinusoidal waveforms is also described. The paper shows that optimum points are predictable for both core size and winding configuration, e.g. use of foil or stranded wire, and the optimum thickness of a foil or the optimum diameter of a copper wire. 2. VISUAL BASIC VS. OBJECT-ORIENTED PROGRAMMING As stated in the introduction, Microsoft Visual Basic was chosen as the development tool for the graphical front-end of the CAD package. It provides an integrated environment for developing programs for Microsoft Windows and Windows for Workgroups. The adoption of Windows ensured that any user already familiar with it would be
Figure 1: Sample design screen from package.
Choose Transformer Application
User
1 Main Process
Access DB Core Data
Transformer Specifications 2 Enter Specifications
Core Type
Application
Area Product
3 Display Matching Cores
Figure 2: Data flow diagram of core selection process. In this screen, the various parameters for a forward converter are entered, and using the formulæ for the various factors (waveform, power, etc.), the Ap value (product of the window area, Wa, and the cross-sectional area, Ac) is calculated. When this value is obtained, the user can then click on the core type of his or her choice, and a corresponding database of core types is opened from which cores with an Ap value slightly greater than the calculated one are automatically selected. This detailed design is discussed with an example in the next section. All core data is accessible using the sophisticated database storage and retrieval capabilities of the DataControl VBX which acts as an intermediary between the CAD package and the Microsoft Access database engine. This engine is supplied with Visual Basic and eliminates the need for the user to buy a copy of Access to run the design package. A number of different core database files are supplied with the package, e.g. one for E-E cores, pot cores etc. If the user is not satisfied with the properties of the various cores suggested by the package, a custom core may be added with user-defined properties to suit that particular application, and this can be saved to the database file for use in future applications.
k p = 1 − D = 1 − 0.66& = 0.58
The output power of the transformer is Po = (8+1)10 = 90 W. A diode forward voltage of 1 V is assumed. This then yields a total VA rating of ⎛ 1
∑ VA = ⎜⎜ ηk ⎝
+
pp
Specifications
Output: 8 V, 10 A Input: 12-36 V Frequency: 25 kHz Temperature Rise, ∆T: 25 °C Efficiency, η: 90%
Assuming equal turns in both the primary and secondary windings, the duty cycle, D, is equal to 8 V/12 V, or 0.66& . The waveform factor, K, relates the RMS value of the voltage waveform, V, to the other parameters in the basic transformer equation
where f is the frequency of operation, N is the number of turns in a winding, Bm is the maximum operating flux density, and Ac is the cross-sectional area of the core.
Adding an extra ∑ VA = 344 VA .
D(1 − D)
5%
for
the
reset
winding
=
⎛ ∑ VA × 104 ⎞ ⎟ A p = ⎜⎜ ⎟ ⎝ KB m fK u K t ∆T ⎠
1.14
(5)
The window utilisation factor, Ku, is assumed to be 0.4, and the temperature factor, Kt, is set to 50 A/cm1.5/°C0.5 [1], [2]. Assuming a ferrite core material, Bm is limited to 0.2 T. In this example, Ap is found to be ⎞ ⎛ 344 × 104 ⎟ A p = ⎜⎜ ⎟ . )( 0.2)( 25 × 103 )( 0.4 )(50) 50 ⎠ ⎝ ( 212
1.14
= 2.58 cm 4
This corresponds to the value calculated by the design package as shown in Figure 1. The Siemens ETD 39 as suggested by the package is an E-core with an Ap value near to this and has the following specifications: Material: Mn-Zn N27 Ac: 1.25 cm2 Wa: 2.34 cm2 Ap: 2.93 cm4
ETD 39
4. MINIMISING LOSSES
Figure 3 is a graphical illustration of how the different losses vary with the number of winding turns [2]. The dotted line represents the core loss, Pfe, the dashed line is the winding (copper) loss, Pcu, and the full line is the sum of these two losses, PL. Reference [2] shows that the losses are of the form
PL = Pcu + Pfe = aN2 +
b N2
c
Losses
(6)
(
)
0.66& 1 − 0.66&
= 212 .
(2)
The power factor, kp, for the input and output windings is given by [1]
e P
P
1
gives
Ap can be calculated from the formula given below [1]:
K can be defined for any shape waveform. The waveform factor for a forward converter was established in [1] as 1
(4)
(1)
V = KfNB m A c
K=
1 ⎞ ⎟ Po k ps ⎟⎠
. ⎛ 172 ⎞ . ⎟ 90 = 327 VA =⎜ + 172 ⎝ 0.9 ⎠
Example: Forward Converter Reference [1] features a design example for a forward converter application with the specifications listed below.
(3)
L
d cu
P
fe
Turns, N
Figure 3: Relationship between losses and turns.
The partial derivative of (6) with respect to N may be taken and set to zero to yield the conditions at which the losses are minimised. In this way, the optimum number of turns is found [as shown in (7)].
4
h
d
0
As can be seen in Figure 3, the copper losses increase and the core losses decrease with increasing N since Pcu ∝ N2 and Pfe ∝ 1 N 2 . Also, since V and f are usually fixed, it can be seen from the basic transformer equation (1) that the maximum flux density, Bm, is inversely proportional to N. The point marked e on the graph corresponds to a core operating at low frequencies. To shift a design towards the optimum point d where the losses are at a minimum, the number of turns must be decreased with a corresponding increase in the maximum flux density. This is not feasible as Bm would then be greater than the saturation flux density, Bsat. Starting at point c (core at high frequencies), optimisation may be achieved by increasing the number of turns and simultaneously reducing Bm. The conventional formula given in (5) is based on the assumption that the total losses are equal to twice the copper losses. This is acceptable at point e in Figure 3, but at point c, the total losses would be underestimated and a new formula which is based on the optimum point d is needed. Such a formula is given in [2]:
(a)
(8)
where α is the coefficient of frequency in the core loss formula Pfe = mK c f α B m
2
W/kg
(9)
135K c with Kc as given in equation (9) for m
in kg. For example, for an Mn-Zn ferrite, Kc = 1.9 × 10-3 and α = 1.24 giving Kb = 1.97 for Ap in m4. Normally Ap is calculated in cm4 so Kb = 19.7.
As illustrated in Figure 4 (a), a layer of insulated round wires has an equivalent foil conductor height, h = 0.834d × d S [3], where d is the diameter of the wire (excluding insulation), and S is the centre-to-centre spacing of the wires. The optimum thickness of foil may now be calculated as follows. Consider a pulse waveform as in Figure 4 (b). This is representative of the current in a push-pull winding. Io is related to the dc output current; for a 1:1 turns ratio, it is equal to the dc output current for a 100% duty cycle. The Fourier Series of i is
3T/2
(b)
2 P = RdcI dc + Rac1 I12 + Rac 3 I 32 +K 2 = RdcI dc +
∞
∑ Rac
n = 1, odd
n
(11)
I 2n
Racn is the ac resistance due to the nth harmonic, and is given by Racn = k pn Rdc , where k pn is the proximity effect factor due to the nth harmonic [5]. Thus ∞
2 P = Rdc Idc + Rdc
∑ k p In2
n = 1, odd
(12)
n
2 but P = Reff I rms giving
R eff = R dc
2 + I dc
1 4 = + 2 2 π
∞
∑ kp
n =1, odd
n
I 2n
I 2rms
(13)
1 ⎡ ⎤ ⎢ k p1 + 9 k p3 +K⎥ ⎣ ⎦
The pulse in Figure 4 (b) is an ideal case. Normally there would be a rise time and fall time associated with the waveform so that a finite number of harmonics are required. Typically, the upper limit on the number of harmonics is 35 tr %
(14)
where tr is the percentage rise time. For example, a 2.5% rise time would give N = 13. Define R δ as the dc resistance of a foil of thickness δ o , where δ o is the skin depth at the fundamental frequency of the pulsed waveform. thickness d and
Rdc is the dc resistance of a foil of
R eff
Rδ d R R = = ∆ ⇒ eff = dc δo ∆ Rdc Rδ
(15)
The ratio Reff Rδ is given the name k r ; for a given
i( t ) = I dc + I$ 1Sin( ωt ) + I$ 3Sin( 3ωt )+K
(10)
I o 2I o ⎡ 1 ⎤ Sin( ωt ) + Sin( 3ωt )+K⎥ + π ⎢⎣ 2 3 ⎦
The RMS value of current is I rms =
T
The total power loss is P = Reff I 2rms which is made up of the dc component and the harmonics
N=
5. OPTIMISING FOIL VS. ROUND CONDUCTOR
T/2
Figure 4: (a) Layer of round wires and equivalent foil height, (b) Current waveform for push-pull converter.
1.33
⎡ ⎤ ∑ VA Ap = ⎢ ⎥ 1− α 2 ∆ KK f K K T ⎢⎣ ⎥⎦ b u t
=
o
(7)
b a
and K b = 1
I
S
∂PL b 2b = 2aN − 3 = 0 ⇒ aN 2 = 2 ∂N N N ∴N =
i
waveform, Rδ is constant and k r is the ac resistance factor for a foil of thickness d.
Io 2
.
Evidently, a plot of k r
versus ∆, the foil thickness to skin depth ratio, has the same shape as a plot of Reff versus d.
The x-axis is increasing foil thickness. For ∆ < ∆ opt , the dc
16
resistance decreases as the thickness increases; however for ∆ > ∆ opt , the ac effects are greater than the effect of
14
(16)
The effective ac resistance of a foil of thickness d is (17)
R eff = k r ∆R dc
⎞ ⎟ ⎟ 0.5 4 ⎟ = ∆ + π2 ⎟ ⎟ ⎠
N
∑
n =1, odd
kpn 2
n ∆
⎡ Sinh( 2 n ∆ ) + Sin( 2 n ∆ ) ⎤ ⎢ ⎥ + ⎢ Cosh( 2 n ∆ ) − Cos( 2 n ∆ ) ⎥ n ∆⎢ ⎥ 2 − − p Sinh( n Sin( n ( ) ) ) ∆ ∆ 2 1 ⎢ ⎥ ⎢ ⎥ 3 Cosh( n ∆ ) + Cos( n ∆ ) ⎦ ⎣
(18)
∆n =
πnfµ r µ oσ
N
∑
n =1, odd
n
(19)
1 ⎡ Sinh( 2 n ∆ ) + Sin( 2 n ∆ ) ⎢ + 3 n 2 ⎢⎣ Cosh( 2 n ∆ ) − Cos( 2 n ∆ )
2( p − 1) Sinh( n ∆ ) − Sin( n ∆ ) ⎤ 0.5 ⎥+ 3 Cosh( n ∆ ) + Cos( n ∆ ) ⎥⎦ ∆
(21)
Regression analysis shows that the following approximation will yield the optimum value for ∆: 1
(22)
⎛N ⎞ ⎛ p2 ⎞ ⎟ ⎜ + 1⎟ ⎜ ⎝2 ⎠ ⎝ 10.56 ⎠
Example: Push-Pull Converter
p = 6, tr = 2.5%, f = 50 kHz, N =
35 35 = = 14 . t r % 2.5
Choose N = 13, since N is odd. From the graph of kr versus ∆ for p = 6 and N = 13 k ropt = 312 . and ∆ opt = 0.43 δo =
66 f
=
p=6
4
p=5
2
kr
opt
∆
opt
66 50 × 103
= 0.295 mm
p=4 p=3 p=2 p=1 1
Figure 6: Resistance factor versus (foil thickness)/(skin depth at fundamental frequency).
Reff = k r ∆Rdc = 312 . × 0.43Rdc = 134 . Rdc
CONCLUSIONS
This paper presents a magnetic component design package which has been developed as an application for the Microsoft Windows environment. An analysis of the transformer equations being implemented in the package is given, and the design procedure is examined. A theoretical example has been compared with an actual computerised implementation of the core selection process. The calculation of the optimum point for minimisation of core and winding losses is illustrated, and also the choice of foil conductors as opposed to round wire for the windings is discussed. A method for obtaining the optimum thickness of a foil conductor is given along with a design example. ACKNOWLEDGEMENTS
kr is plotted as a function of ∆, with p and N specified.
4
6
A 0.13 mm × 30 mm foil has the same copper area as a 2.24 mm diameter of bare copper wire, yielding Reff = 2.15Rdc.
(20)
2
∆ opt =
p=7
and the ac resistance is
δo
d d = n = n∆ δn δo
4 π2
8
d opt = ∆ optδ o = 0.43 × 0.295 = 013 . mm
k r is now given by
kr =
r
The optimum foil thickness is equal to
where p is the number of layers of foil. The skin depth at the nth harmonic is =
k
p=8
∆
k pn is given by Dowell's formula [5]
1
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
⎛ 4 N kpn ⎜ 0.5 + 2 ∑ π n =1, odd n 2 ⎜ kr = ⎜ ∆ ⎜ ⎜ ⎝
δn =
N = 13
0
From equation (12)
k pn =
p=9
12
increased thickness. The minimum ac resistance is given at ∆ opt and the optimum thickness is
dopt = ∆ opt .δ o
p=10
This work has been funded by Power Electronics Ireland. REFERENCES
[1] Hurley, W.G., O’Brien, M.H.: “Issues Related to the Computer Aided Design (CAD) of High Frequency Transformers”, 26th Universities Power Engineering Conference, Brighton Polytechnic, September 1991. [2] Hurley, W.G., Woelfle, W.: “Optimizing the Design of Transformers at High Frequencies”, 27th Universities Power Engineering Conference, University of Bath, September 1992. [3] Carsten, B.: “High Frequency Conductor Losses in Switchmode Magnetics”, HFPC Proceedings, May 1986. [4] Judd, F.F., Kressler, D.R.: “Design Optimization of Small Low-Frequency Power Transformers”, IEEE Transactions on Magnetics, Vol. MAG-13 No. 4, July 1977. [5] Dowell, P.L.: “Effects of Eddy Currents in Transformer Windings”, IEEE Proceedings, Vol. 113 No. 8, August 1966.
